The mutual information of two random variables z and J with joint probabilities {7rij} is commonly used in learning Bayesian nets as well as in many other fields. The chances 7rij are usually estimated by the empirical sampling frequency nij In leading to a point estimate J(nij In) for the mutual information. To answer questions like "is J (nij In) consistent with zero?" or "what is the probability that the true mutual information is much larger than the point estimate?"

We study online learning in Boolean domains using kernels which capture feature expansions equivalent to using conjunctions over basic features. We demonstrate a tradeoff between the computational efficiency with which these kernels can be computed and the generalization ability of the resulting classifier. We first describe several kernel functions which capture either limited forms of conjunctions or all conjunctions. We show that these kernels can be used to efficiently run the Perceptron algorithm over an exponential number of conjunctions; however we also prove that using such kernels the Perceptron algorithm can make an exponential number of mistakes even when learning simple functions. We also consider an analogous use of kernel functions to run the multiplicative-update Winnow algorithm over an expanded feature space of exponentially many conjunctions. While known upper bounds imply that Winnow can learn DNF formulae with a polynomial mistake bound in this setting, we prove that it is computationally hard to simulate Winnow's behavior for learning DNF over such a feature set, and thus that such kernel functions for Winnow are not efficiently computable.

We introduce the notion of kernel-alignment, a measure of similarity between two kernel functions or between a kernel and a target function. This quantity captures the degree of agreement between a kernel and a given learning task, and has very natural interpretations in machine learning, leading also to simple algorithms for model selection and learning. We analyse its theoretical properties, proving that it is sharply concentrated around its expected value, and we discuss its relation with other standard measures of performance. Finally we describe some of the algorithms that can be obtained within this framework, giving experimental results showing that adapting the kernel to improve alignment on the labelled data significantly increases the alignment on the test set, giving improved classification accuracy. Hence, the approach provides a principled method of performing transduction.

In this paper we show that online algorithms for classification and regression can be naturally used to obtain hypotheses with good datadependent tail bounds on their risk. Our results are proven without requiring complicated concentration-of-measure arguments and they hold for arbitrary online learning algorithms. Furthermore, when applied to concrete online algorithms, our results yield tail bounds that in many cases are comparable or better than the best known bounds.

We consider noisy Euclidean traveling salesman problems in the plane, which are random combinatorial problems with underlying structure. Gibbs sampling is used to compute average trajectories, which estimate the underlying structure common to all instances. This procedure requires identifying the exact relationship between permutations and tours. In a learning setting, the average trajectory is used as a model to construct solutions to new instances sampled from the same source. Experimental results show that the average trajectory can in fact estimate the underlying structure and that overfitting effects occur if the trajectory adapts too closely to a single instance.

Singularities are ubiquitous in the parameter space of hierarchical models such as multilayer perceptrons. At singularities, the Fisher information matrix degenerates, and the Cramer-Rao paradigm does no more hold, implying that the classical model selection theory such as AIC and MDL cannot be applied. It is important to study the relation between the generalization error and the training error at singularities. The present paper demonstrates a method of analyzing these errors both for the maximum likelihood estimator and the Bayesian predictive distribution in terms of Gaussian random fields, by using simple models. 1 Introduction A neural network is specified by a number of parameters which are synaptic weights and biases. Learning takes place by modifying these parameters from observed input-output examples.

Asymmetric lateral connections are one possible mechanism that can account for the direction selectivity of cortical neurons. We present a mathematical analysis for a class of these models. Contrasting with earlier theoretical work that has relied on methods from linear systems theory, we study the network's nonlinear dynamic properties that arise when the threshold nonlinearity of the neurons is taken into account. We show that such networks have stimulus-locked traveling pulse solutions that are appropriate for modeling the responses of direction selective cortical neurons. In addition, our analysis shows that outside a certain regime of stimulus speeds the stability of this solutions breaks down giving rise to another class of solutions that are characterized by specific spatiotemporal periodicity. This predicts that if direction selectivity in the cortex is mainly achieved by asymmetric lateral connections lurching activity waves might be observable in ensembles of direction selective cortical neurons within appropriate regimes of the stimulus speed.

Inferior temporal cortex (IT) neurons have large receptive fields when a single effective object stimulus is shown against a blank background, but have much smaller receptive fields when the object is placed in a natural scene. Thus, translation invariant object recognition is reduced in natural scenes, and this may help object selection. We describe a model which accounts for this by competition within an attractor in which the neurons are tuned to different objects in the scene, and the fovea has a higher cortical magnification factor than the peripheral visual field. Furthermore, we show that top-down object bias can increase the receptive field size, facilitating object search in complex visual scenes, and providing a model of object-based attention. The model leads to the prediction that introduction of a second object into a scene with blank background will reduce the receptive field size to values that depend on the closeness of the second object to the target stimulus. We suggest that mechanisms of this type enable the output of IT to be primarily about one object, so that the areas that receive from IT can select the object as a potential target for action.

Hebbian learning rules are generally formulated as static rules.

Population codes often rely on the tuning of the mean responses to the stimulus parameters. However, this information can be greatly suppressed by long range correlations. Here we study the efficiency of coding information in the second order statistics of the population responses. We show that the Fisher Information of this system grows linearly with the size of the system. We propose a bilinear readout model for extracting information from correlation codes, and evaluate its performance in discrimination and estimation tasks. It is shown that the main source of information in this system is the stimulus dependence of the variances of the single neuron responses.